What Number Must Be Added To Complete The Square

Completing the square is a fundamental algebraic technique used to solve quadratic equations, rewrite quadratic expressions in vertex form, and simplify various mathematical problems. The core idea involves transforming a quadratic expression into a perfect square trinomial. This article will delve into the process of determining what number must be added to complete the square, providing a comprehensive understanding with examples, explanations, and applications.

Understanding Completing the Square

Completing the square is a method used to rewrite a quadratic expression of the form ax² + bx + c into the form a(x + h)² + k, where (h, k) represents the vertex of the parabola. The process involves manipulating the quadratic expression to create a perfect square trinomial, which can then be factored into a binomial squared.

The Perfect Square Trinomial

A perfect square trinomial is a trinomial that can be factored into the square of a binomial. The general form of a perfect square trinomial is:

a² + 2ab + b² = (a + b)²

or

a² - 2ab + b² = (a - b)²

To complete the square, we aim to transform a given quadratic expression into one of these forms.

Steps to Complete the Square

The process of completing the square involves several key steps. Here’s a detailed breakdown:

1. Ensure the Coefficient of x² is 1: If the coefficient of x² (i.e., a) is not 1, divide the entire expression by a. This simplifies the process of finding the term needed to complete the square.
2. Identify the Coefficient of x: Find the coefficient of the x term (i.e., b).
3. Calculate the Value to Add: Divide the coefficient of x by 2, and then square the result. This value is the number that must be added to complete the square.
4. Add and Subtract the Value: Add and subtract the value calculated in step 3 inside the expression. This maintains the expression’s original value while allowing us to create a perfect square trinomial.
5. Factor the Perfect Square Trinomial: Factor the perfect square trinomial into the square of a binomial.
6. Simplify the Expression: Combine the remaining terms to obtain the completed square form.

Step-by-Step Example

Let's consider the quadratic expression x² + 6x + 5 and complete the square:

1. Coefficient of x² is 1: The coefficient of x² is already 1, so no division is needed.
2. Identify the Coefficient of x: The coefficient of x is 6.
3. Calculate the Value to Add: Divide the coefficient of x by 2 and square the result: (b/2)² = (6/2)² = (3)² = 9
4. Add and Subtract the Value: Add and subtract 9 inside the expression: x² + 6x + 9 - 9 + 5
5. Factor the Perfect Square Trinomial: Factor x² + 6x + 9 into (x + 3)²: (x + 3)² - 9 + 5
6. Simplify the Expression: Combine the remaining terms: (x + 3)² - 4

The completed square form of x² + 6x + 5 is (x + 3)² - 4.

Detailed Explanation of Each Step

To fully grasp the concept, let's break down each step with additional insights and examples.

1. Ensure the Coefficient of x² is 1

When the coefficient of x² is not 1, it complicates the process of completing the square. Dividing the entire expression by the coefficient ensures that the leading term is x², making it easier to identify and manipulate the x term.

Example:

Consider the expression 2x² + 8x + 6. To make the coefficient of x² equal to 1, divide the entire expression by 2:

2x² + 8x + 6 = 2(x² + 4x + 3)

Now, we complete the square for the expression inside the parentheses, x² + 4x + 3.

2. Identify the Coefficient of x

The coefficient of x (denoted as b in the general form ax² + bx + c) is crucial for determining the value needed to complete the square. This value is used in the formula (b/2)², which is the cornerstone of the completing the square method.

Example:

In the expression x² + 10x + 20, the coefficient of x is 10.

3. Calculate the Value to Add

The value to add is calculated by dividing the coefficient of x by 2 and then squaring the result. This value ensures that when added to the expression, a perfect square trinomial is formed.

Formula:

Value to add = (b/2)²

Explanation:

The perfect square trinomial has the form a² + 2ab + b². In our case, a is x, and we want to find a value that, when added, completes the square. The x term in the expression is 2ab, which corresponds to bx in the general form. Therefore, 2ab = bx, and since a = x, we have 2xb = bx, which implies that b = b/2. To complete the square, we need to add b², which is (b/2)².

Example:

For the expression x² + 10x + 20, the coefficient of x is 10. The value to add is:

Value to add = (10/2)² = (5)² = 25

4. Add and Subtract the Value

Adding and subtracting the same value inside the expression does not change the expression's overall value but allows us to manipulate it into the desired form. This step is crucial for maintaining the integrity of the equation while creating a perfect square trinomial.

Example:

For the expression x² + 10x + 20, we add and subtract 25:

x² + 10x + 25 - 25 + 20

5. Factor the Perfect Square Trinomial

The perfect square trinomial can now be factored into the square of a binomial. This is the key step that transforms the expression into the completed square form.

Factoring:

x² + 2bx + b² = (x + b)²

Example:

In the expression x² + 10x + 25 - 25 + 20, the perfect square trinomial x² + 10x + 25 can be factored into (x + 5)².

So, we have:

(x + 5)² - 25 + 20

6. Simplify the Expression

Finally, simplify the expression by combining the remaining terms to obtain the completed square form, a(x + h)² + k.

Example:

In the expression (x + 5)² - 25 + 20, combine -25 and 20:

(x + 5)² - 5

The completed square form of x² + 10x + 20 is (x + 5)² - 5.

Examples with Different Coefficients

Let's go through several examples to solidify the understanding of completing the square.

Example 1: x² - 8x + 12

1. Coefficient of x² is 1: Already 1.
2. Coefficient of x: -8
3. Value to Add: ((-8)/2)² = (-4)² = 16
4. Add and Subtract the Value: x² - 8x + 16 - 16 + 12
5. Factor the Perfect Square Trinomial: (x - 4)² - 16 + 12
6. Simplify the Expression: (x - 4)² - 4

Completed square form: (x - 4)² - 4

Example 2: 3x² + 12x + 9

1. Coefficient of x² is 1: Divide by 3: 3(x² + 4x + 3)
2. Coefficient of x: 4
3. Value to Add: (4/2)² = (2)² = 4
4. Add and Subtract the Value: 3(x² + 4x + 4 - 4 + 3)
5. Factor the Perfect Square Trinomial: 3[(x + 2)² - 4 + 3]
6. Simplify the Expression: 3[(x + 2)² - 1] = 3(x + 2)² - 3

Completed square form: 3(x + 2)² - 3

Example 3: x² + 5x + 4

1. Coefficient of x² is 1: Already 1.
2. Coefficient of x: 5
3. Value to Add: (5/2)² = 25/4
4. Add and Subtract the Value: x² + 5x + 25/4 - 25/4 + 4
5. Factor the Perfect Square Trinomial: (x + 5/2)² - 25/4 + 16/4
6. Simplify the Expression: (x + 5/2)² - 9/4

Completed square form: (x + 5/2)² - 9/4

Applications of Completing the Square

Completing the square is not just an algebraic exercise; it has several important applications in mathematics and related fields.

1. Solving Quadratic Equations

Completing the square can be used to solve quadratic equations of the form ax² + bx + c = 0. By rewriting the equation in the completed square form, we can easily isolate x and find the solutions.

Example:

Solve the equation x² + 6x + 5 = 0 by completing the square:

1. Complete the square: (x + 3)² - 4 = 0
2. Isolate the squared term: (x + 3)² = 4
3. Take the square root of both sides: x + 3 = ±2
4. Solve for x:
   • x = -3 + 2 = -1
   • x = -3 - 2 = -5

The solutions are x = -1 and x = -5.

2. Finding the Vertex of a Parabola

The completed square form a(x + h)² + k directly gives the vertex of the parabola represented by the quadratic equation. The vertex is the point (h, k), which is either the minimum or maximum point of the parabola.

Example:

Find the vertex of the parabola y = x² + 4x + 7:

1. Complete the square: y = (x + 2)² + 3
2. The vertex is (-2, 3).

3. Rewriting Quadratic Functions in Vertex Form

Rewriting a quadratic function in vertex form makes it easier to analyze and graph the function. The vertex form provides immediate information about the parabola's vertex, axis of symmetry, and direction of opening.

Example:

Rewrite y = 2x² - 8x + 10 in vertex form:

1. Factor out the coefficient of x²: y = 2(x² - 4x + 5)
2. Complete the square inside the parentheses: y = 2[(x - 2)² + 1]
3. Distribute the 2: y = 2(x - 2)² + 2

The vertex form is y = 2(x - 2)² + 2.

4. Integration in Calculus

Completing the square can simplify integrals involving quadratic expressions in the denominator. By completing the square, the integral can be transformed into a standard form that can be easily evaluated.

Example:

Evaluate the integral ∫ dx / (x² + 2x + 2):

1. Complete the square in the denominator: x² + 2x + 2 = (x + 1)² + 1
2. Rewrite the integral: ∫ dx / [(x + 1)² + 1]
3. Use the substitution u = x + 1, so du = dx: ∫ du / (u² + 1)
4. The integral is now in a standard form: arctan(u) + C
5. Substitute back x + 1 for u: arctan(x + 1) + C

The integral evaluates to arctan(x + 1) + C.

Common Mistakes to Avoid

While completing the square is a straightforward process, there are common mistakes that students and practitioners often make. Avoiding these mistakes can ensure accurate results.

1. Forgetting to Divide by the Coefficient of x²: When the coefficient of x² is not 1, failing to divide the entire expression by this coefficient before completing the square is a common error.
2. Incorrectly Calculating the Value to Add: Making mistakes in calculating (b/2)² can lead to an incorrect perfect square trinomial.
3. Forgetting to Add and Subtract the Value: Adding the value to complete the square without also subtracting it changes the expression’s value.
4. Incorrectly Factoring the Perfect Square Trinomial: Factoring the perfect square trinomial incorrectly can lead to an incorrect completed square form.
5. Errors in Simplifying the Expression: Making arithmetic errors while simplifying the expression after factoring the perfect square trinomial is a common mistake.

Advanced Techniques and Applications

Completing the square can also be extended to more complex problems and scenarios.

1. Completing the Square with Complex Numbers

The process of completing the square can be applied to quadratic expressions involving complex numbers. This is particularly useful in solving quadratic equations with complex roots.

Example:

Solve z² + 2iz - 1 = 0, where z is a complex number:

1. Complete the square: (z + i)² - i² - 1 = 0
2. Simplify: (z + i)² + 1 - 1 = 0
3. (z + i)² = 0
4. z + i = 0
5. z = -i

The solution is z = -i.

2. Completing the Square in Multivariable Calculus

In multivariable calculus, completing the square can be used to simplify expressions in multiple variables. This is often used in optimization problems and in evaluating certain types of integrals.

Example:

Consider the function f(x, y) = x² + 2x + y² - 4y + 5. To find the minimum value, complete the square for both x and y terms:

1. Complete the square for x: (x + 1)² - 1
2. Complete the square for y: (y - 2)² - 4
3. Rewrite the function: f(x, y) = (x + 1)² - 1 + (y - 2)² - 4 + 5
4. Simplify: f(x, y) = (x + 1)² + (y - 2)²

The minimum value of the function is 0, which occurs when x = -1 and y = 2.

3. Applications in Physics and Engineering

Completing the square has applications in various fields of physics and engineering, such as analyzing circuits, solving differential equations, and optimizing system parameters.

Example:

In circuit analysis, the voltage across an inductor and a resistor in series can be described by a quadratic equation. Completing the square can help determine the time at which the voltage reaches its maximum or minimum value.

Conclusion

Completing the square is a powerful and versatile algebraic technique with numerous applications in mathematics, science, and engineering. By understanding the steps involved and practicing with various examples, one can master this technique and apply it effectively to solve a wide range of problems. Whether it's solving quadratic equations, finding the vertex of a parabola, or simplifying complex integrals, completing the square is an essential tool in the problem-solving arsenal. The number that must be added to complete the square is determined by dividing the coefficient of the x term by 2 and squaring the result, a process that transforms a quadratic expression into a perfect square trinomial, facilitating simplification and analysis.